Try "every 2 weeks" instead. Start by having a due date on a Monday.

I tried that, and it works, but when I postpone the task, the due date changes - and doesn't get reset when the task is regenerated.

Don't postpone ;-)
Sorry, missed that, I never use postpone, I just let the task become overdue. But then again, I suppose the new instance will be due two weeks later after completion, not necessarily on a Monday. Hmm, difficult...
But if you can postpone the task, is it really due on that Monday? Maybe it's OK to repeat after 2 weeks?

Well, the thing is - I have approx. two hours of no-brain work every week (checking different client's backup every two weeks, security patching, etc.). I do that every monday morning between 8 and 10. I never schedule meetings in that time, so I know that I will get it done. But as always there are exceptions, this week I had to do a meeting so I didn't get around to performing the tasks until today. Now the tasks have a due date for Tuesday in two weeks. See the problem?

I see your point. The problem is the postponing, postponing it for 1 day moves the whole series to Tuesdays, with no return to Mondays.

If you don't postpone but instead let it become overdue (still on top of lists if sorted by due date) it will work better. A new instance will be generated automatically at midnight Monday-Tuesday, due Monday two weeks from now. The old instance is overdue and completing it will not generate a new task on a Tuesday two weeks from now.